[Paper Review] Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect

Piermarco Cannarsa, Valerio Lucarini+3 more|arXiv (Cornell University)|Nov 28, 2022

Earth Systems and Cosmic Evolution56 references4 citations

TL;DR

This paper analyzes a two-layer energy balance model that captures the greenhouse effect through atmospheric absorptivity $\varepsilon_a$. It proves that solutions exist globally if $\varepsilon_a \in (0,2)$, blow up if $\varepsilon_a > 2$, and converge to equilibrium states for $\varepsilon_a \in (0,2)$, with surface temperature monotonically increasing in $\varepsilon_a$, mathematically confirming the greenhouse effect.

ABSTRACT

We study a two-layer energy balance model, that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being captured by the other layer, and the effect of all non radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted $ε_a$. The value of $ε_a$ depends critically on greenhouse gases: increasing concentrations of $CO_2$ and $CH_4$ lead to a more opaque atmosphere with higher values of $ε_a$. First we prove that global existence of solutions of the system holds if and only if $ε_a \in (0, 2)$, and blow up in finite time occurs if $ε_a > 2$. (Note that the physical range of values for $ε_a$ is $(0, 1]$.) Next, we explain the long time dynamics for $ε_a \in (0, 2)$, and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove in particular that the surface temperature increases with respect to $ε_a$. This is the key mathematical manifestation of the greenhouse effect.

Motivation & Objective

• To rigorously analyze the long-time behavior of a two-layer energy balance model with coupled surface and atmospheric temperatures.
• To determine the conditions under which solutions exist globally or blow up in finite time.
• To establish the dependence of equilibrium states on the atmospheric absorptivity $\varepsilon_a$, a key parameter for the greenhouse effect.
• To mathematically confirm the monotonic increase of surface temperature with increasing $\varepsilon_a$, representing the greenhouse effect.

Proposed method

• Formulation of a two-layer energy balance model with coupled evolution equations for surface and atmospheric temperatures.
• Incorporation of infrared radiation exchange and non-radiative energy fluxes, parameterized by atmospheric absorptivity $\varepsilon_a$.
• Use of energy balance principles and Stefan-Boltzmann radiation laws to model outgoing longwave radiation.
• Application of dynamical systems theory to study asymptotic behavior and convergence to equilibrium points.
• Analysis of the equilibrium equation $\Phi(T_s) = q\beta_s(T_s)$ using convexity and Rolle’s Theorem to bound the number of solutions.
• Numerical and analytical investigation of the function $\Phi$, including its derivatives, to determine the number of equilibrium states for varying $\varepsilon_a$.

Experimental results

Research questions

• RQ1 Under what conditions on $\varepsilon_a$ do solutions to the two-layer energy balance model exist globally in time?
• RQ2 What is the long-time behavior of the system when $\varepsilon_a \in (0,2)$, and do all solutions converge to equilibrium points?
• RQ3 How does the number of equilibrium solutions depend on the value of $\varepsilon_a$, particularly near the critical threshold $\varepsilon_a = 2$?
• RQ4 Is the surface temperature monotonically increasing with respect to $\varepsilon_a$, and does this provide a mathematical basis for the greenhouse effect?
• RQ5 Can the model support multiple equilibrium states, and if so, under what parameter conditions?

Key findings

• Solutions to the two-layer energy balance model exist globally in time if and only if $\varepsilon_a \in (0,2)$, and blow up in finite time if $\varepsilon_a > 2$, despite the physical range being $\varepsilon_a \in (0,1]$.
• For all $\varepsilon_a \in (0,2)$, all solutions converge to some equilibrium point, indicating stable long-term behavior under physically relevant conditions.
• The surface temperature increases monotonically with respect to $\varepsilon_a$, providing a rigorous mathematical confirmation of the greenhouse effect.
• The number of equilibrium solutions is at most six, and for physically relevant parameters, numerical evidence suggests at most three equilibria exist, even near $\varepsilon_a \to 2^-$.
• When $\varepsilon_a \in (0,1]$, the function $\Phi$ is strictly convex and strictly increasing, ensuring at most three equilibrium solutions.
• For $\varepsilon_a \in (\varepsilon_{a,0}, 2)$ with $\varepsilon_{a,0} \approx 1.99$, $\Phi$ exhibits a convex-concave-convex shape, allowing at most five equilibrium solutions, but numerical results suggest only up to three are physically relevant.